How to verify Mandrill custom sending domain without a mail receiving client?

I was having the same problem earlier. I had a web application running on a server. So I configured incoming mails on mandrill by adding some MX records, then creating a route for the same domain for a random address which points to one of the POST endpoints on my web application.

So, whenever someone sends an email to that address, I get a POST request on the web endpoint which I put in the control panel. Mandrill gives a confirmation url in the email body. So, by inspecting the logs, we can directly copy the url and paste it in the browser and the domain is verified.

You can delete the incoming domain in mandrill afterwards if you don't need it.